On Unit Distances in a Convex Polygon

Details On Unit Distances in a Convex Polygon On Unit Distances in a Convex Polygon

2010-09-12

arxiv.org/abs/1009.2216v1

Computer Science

Computational Geometry

Scientific paper

For any convex quadrilateral, the sum of the lengths of the diagonals is greater than the corresponding sum of a pair of opposite sides, and all four of its interior angles cannot be simultaneously acute. In this article, we use these two properties to estimate the number of unit distance edges in convex n-gons and we: (i) exhibit three large groups of cycles formed by unit distance edges that are forbidden in convex n-gons, (ii) prove that the maximum number of unit distances is at most $n \log_2 n + 4n$, thereby improving the best known result by a factor of $2\pi$, and (iii) we show that if we only use these two properties then we will not be able to further improve this bound by more than a factor of four.

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